Structural Health Monitoring

Structural
Health Monitoring
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WaveFormRevealer: An analytical framework and predictive tool for the simulation of multi-modal
guided wave propagation and interaction with damage
Yanfeng Shen and Victor Giurgiutiu
Structural Health Monitoring published online 13 May 2014
DOI: 10.1177/1475921714532986
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Original Article
WaveFormRevealer: An analytical
framework and predictive tool for the
simulation of multi-modal guided wave
propagation and interaction with
damage
Structural Health Monitoring
1–21
Ó The Author(s) 2014
Reprints and permissions:
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DOI: 10.1177/1475921714532986
shm.sagepub.com
Yanfeng Shen and Victor Giurgiutiu
Abstract
This article presents the WaveFormRevealer—an analytical framework and predictive tool for the simulation of guided
Lamb wave propagation and interaction with damage. The theory of inserting damage effects into the analytical model is
addressed, including wave transmission, reflection, mode conversion, and nonlinear higher harmonics. The analytical
model is coded into MATLAB, and a graphical user interface (WaveFormRevealer graphical user interface) is developed
to obtain real-time predictive waveforms for various combinations of sensors, structural properties, and damage. In this
article, the main functions of WaveFormRevealer are introduced. Case studies of selective Lamb mode linear and nonlinear interaction with damage are presented. Experimental verifications are carried out. The article finishes with summary and conclusions followed by recommendations for further work.
Keywords
Guided waves, structural health monitoring, damage detection, piezoelectric wafer active sensors, analytical model, nonlinear ultrasonics
Introduction
Guided waves retain a central function in the development of structural health monitoring (SHM) systems
using piezoelectric wafer active sensor (PWAS) principles. The modeling of Lamb waves is challenging,
because Lamb waves propagate in structures with
multi-mode dispersive characteristics. At a certain
value of the plate thickness-frequency product, several
Lamb modes may exist simultaneously, and their phase
velocities vary with frequency.1–3 When Lamb waves
interact with damage, they will be transmitted,
reflected, scattered, and mode converted. Nonlinear
interaction with damage may also exist, and this will
introduce distinctive features like higher harmonics.4–6
These aspects give rise to the complexity of modeling
the interaction between Lamb waves and damage. To
solve such complicated problems, numerical methods
like finite element method (FEM) and boundary element method (BEM) are usually adopted. However, to
ensure the accuracy of simulating high-frequency waves
of short wavelengths, the transient analysis requires
considerably small time step and very fine mesh
(T =Dt; l=lFEM 20 ; 30), which is expensive both in
computational time and computer resources.7,8
Analytical model provides an alternative approach to
attack the same problem with much less cost.9
PWAS transducers are a convenient way of transmitting and receiving guided waves in structures for SHM
applications.3 The analytical model of PWAS-generated
Lamb waves and its tuning effect has been investigated,
and a close-form solution for straight crested guided
Lamb wave was derived by Giurgiutiu.10 Extension of
tuning concepts to 2D analytical models of Lamb waves
generated by finite-dimensional piezoelectric transducers was given in Raghavan and Cesnik.11 These analytical developments facilitate the understanding of
PWAS-coupled Lamb waves for SHM applications.
Department of Mechanical Engineering, University of South Carolina,
Columbia, SC, USA
Corresponding author:
Yanfeng Shen, Department of Mechanical Engineering, University of South
Carolina, Columbia, SC 29208, USA.
Email: [email protected]
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2
Structural Health Monitoring
However, these analytical solutions only applied to
guided wave propagation in pristine structures, whereas
the use of Lamb waves in SHM applications requires
that their interaction with damage also be studied.
After interacting with damage, Lamb waves will carry
damage information resulting in waveforms with special characteristics (phase change, new wave packets
generation through mode conversion, higher harmonic
components, etc.), which need to be investigated for
damage detection.
Several investigators have studied the interaction
between guided waves and damage analytically using
normal-mode expansion and boundary-condition
matching.12–18 Damage interaction coefficients were
derived to quantify the guided wave transmission,
reflection, mode conversion, and scatter at the damage
site. Due to their mathematical complexity, these analytical solutions are restricted to simple damage geometries: notches, holes, and partially through holes.
Extension to more complicated damage geometries has
been attempted through series expansion of the rugged
damage contour. In the generic case of arbitrary-shape
damage, the numerical approaches using space discretization (FEM, BEM) are used due to their convenience,
but on the expense of orders of magnitude increase in
computational time and/or computer resources.
The design of a SHM system requires computationally efficient predictive tools that permit the exploration
of a wide parameter space to identify the optimal combination between the transducers type, size, number,
and guided wave characteristics (mode type, frequency,
and wavelength) to achieve best detection and quantification of a certain damage type. Such parameter space
exploration desiderate can be best achieved with analytical tools which are fast and efficient.
In this article, we describe an analytical approach
based on the one-dimensional (1D) (straight crested)
guided wave propagation analysis. In our study, we
inserted the damage effect into the analytical model by
considering wave transmission, reflection, mode conversion, and higher harmonics components described
through damage interaction coefficients at the damage
site. We do not attempt to derive these damage interaction coefficients here, but assume that they are available
either from literature or from FEM, BEM analysis performed separately in a separate computational module.
This analytical approach was coded into MATLAB
and the WaveFormRevealer (WFR) graphical user
interface (GUI) was developed. The WFR can generate
fast predictions of waveforms resulting from Lamb
wave interaction with damage for arbitrary positioning
of PWAS transmitters and receivers with respect to
damage and with respect to each other. The users may
choose their own excitation signal, PWAS size, structural parameters, and damage description. The current
version of the WFR code is limited to 1D (straight
crested) guided wave propagation; extension of this
approach to two-dimensional (2D) (circular crested)
guided wave propagation will be attempted in the
future.
PWAS Fundamentals
PWAS couple the electrical and mechanical effects
(mechanical strain, Sij ; mechanical stress, Tkl ; electrical
field, Ek ; and electrical displacement, Dj ) through the
tensorial piezoelectric constitutive equations
Sij ¼ sEijkl Tkl þ dkij Ek
Dj ¼ djkl Tkl þ eTjk Ek
ð1Þ
where sEijkl is the mechanical compliance of the material
measured at zero electric field (E ¼ 0), eTjk is the dielectric permittivity measured at zero mechanical stress
(T ¼ 0), and djkl represents the piezoelectric coupling
effect. PWAS utilize the d31 coupling between in-plane
strains, S1 ; S2 , and transverse electric field E3 .
PWAS transducers can be used as both transmitters
and receivers. Their modes of operation are shown in
Figure 1. PWAS can serve several purposes3: (a) highbandwidth strain sensors, (b) high-bandwidth wave
exciters and receivers, (c) resonators, and (d) embedded
modal sensors with the electromechanical (E/M) impedance method. By application types, PWAS transducers
can be used for (a) active sensing of far-field damage
using pulse-echo, pitch-catch, and phased-array methods, (b) active sensing of near-field damage using highfrequency E/M impedance method and thickness gage
mode, and (c) passive sensing of damage-generating
events through detection of low-velocity impacts and
acoustic emission at the tip of advancing cracks
(Figure 1). The main advantage of PWAS over conventional ultrasonic probes is in their small size, lightweight, low profile, and small cost. In spite of their
small size, PWAS are able to replicate many of the
functions performed by conventional ultrasonic probes.
Analytical modeling of Lamb waves
interacting with damage
Analytical modeling of guided Lamb waves
propagation in a pristine structure
One aspect of the difficulties in modeling Lamb wave
propagation is due to their multi-mode feature. WFR
is capable of modeling multi-mode Lamb wave propagation in structures. From Rayleigh–Lamb equation, it
is found that the existence of certain Lamb mode
depends on the plate thickness-frequency product. The
fundamental S0 and A0 modes will always exist, but
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Shen and Giurgiutiu
3
Figure 1. Schematic of PWAS application modes (from Giurgiutiu19): (a) propagating Lamb waves, (b) standing Lamb waves (E/M
impedance), and (c) PWAS phased arrays.
E/M: electromechanical; PWAS: piezoelectric wafer active sensor; AE: acoustic emission.
Transmitter PWAS
Excitation: VT (ω )
Receiver PWAS
Signal: VR ( xr , ω )
x=0
x=xr
T-PWAS
Piezoelectric
transduction
Elec.→Mech.
R-PWAS
Ultrasonic guided waves from T-PWAS
undergo dispersion according to structural
transfer function G ( xr , ω )
Piezoelectric
transduction
Mech.→Elec.
Figure 2. A pitch-catch configuration between a transmitter
PWAS (T-PWAS) and a receiver PWAS.
PWAS: piezoelectric wafer active sensor.
the higher modes will only appear beyond the cutoff
frequencies.1
This section describes how an electrical tone burst
applied to a transmitter PWAS (T-PWAS) transducer
propagates through a structural waveguide to the receiver PWAS (R-PWAS) transducer in pitch-catch mode
(Figure 2).
The propagation takes place through ultrasonic
guided Lamb waves which are generated at the
T-PWAS through piezoelectric transduction and then
captured and converted back into electric signal at the
R-PWAS. Since several Lamb wave modes traveling
with different wave speeds exist simultaneously, the
electrical tone-burst applied on the T-PWAS will
generate several wave packets. These wave packets will
travel independently through the waveguide and will
arrive at different times at the R-PWAS where they are
converted back into electric signals through piezoelectric transduction. The predictive analytical model for
Lamb wave propagation between the T-PWAS and RPWAS is constructed in frequency domain in the following steps (Figure 3(a)).
Step 1. Perform Fourier transform of the time-domain
excitation signal VT ðtÞ to obtain the frequency-domain
~T ðvÞ. For a tone burst, the signal
excitation spectrum, V
~T ðvÞ are shown in
VT ðtÞ and its Fourier transform V
Figure 4.
Step 2. Calculate the frequency-domain structural transfer function Gðxr ; vÞ from T-PWAS to R-PWAS. The
structure transfer function Gðxr ; vÞ is given in equation
(99) in the study by Giurgiutiu,3 page 327, which gives
the in-plane wave strain at the plate surface as
8
NS jS iðjS xvtÞ
at 0 <X
S
ex ðx; tÞ ¼ i
ðsin j aÞ 0 S e
m : S
DS j
j
)
A
X
NA j
iðjA xvtÞ
A
þ
ðsin j aÞ 0 A e
DA j
jA
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ð2Þ
4
Structural Health Monitoring
(a)
(b)
Flow chart for prisne case
guided wave propagaon
Flow chart for guided wave propagaon
with damage interacon
Fourier transform of
excitaon signal
Fourier transform of
excitaon signal
FFT
VT ( t ) → VT (ω )
VT ( t ) → VT (ω )
Calculaon of structural
transfer funcon
Calculaon of structural transfer
funcon up to damage locaon
G ( xr , ω )
G ( xd , ω )
Mulply excitaon with
structural transfer funcon
Mulply excitaon with structural
transfer funcon to the damage locaon
FFT
VD ( xd , ω ) = G ( xd , ω )iVT (ω )
VR ( xr , ω ) = G ( xr , ω )iVT (ω )
Transmission, reﬂecon,
mode conversion, and
higher harmonics
New wave source
VNewSource ( xd , ω )
This new wave source is mulplied with
structural transfer funcon from damage
upto R-PWAS
VR ( xd , xr , ω ) = G ( xr − xd , ω )iVNewSource ( xd , ω )
Perform inverse Fourier transform
{
}
VT ( xr , t ) = IFFT VR ( xr , ω )
Perform inverse Fourier transform
VT ( xd , xr , t ) = IFFT {VR ( xd , xr , ω )}
Figure 3. WaveFormRevealer flow charts: (a) propagation in a pristine structural waveguide and (b) propagation and interaction
with damage at location xd .
where j is the frequency-dependent wavenumber
of each Lamb wave mode and the superscripts S and
A refer to symmetric and antisymmetric Lamb
wave modes. If only the two fundamental modes,
S0 and A0, are present, then Gðxr ; vÞ can be written
as
Gðxr ; vÞ ¼ S ðvÞeij
S
þ AðvÞeij
S
S NS j
S ðvÞ ¼ kPWAS sin j a 0 S ;
D S j NA jA
AðvÞ ¼ kPWAS sin jA a 0 A DA j
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xr
A
xr
ð3Þ
Shen and Giurgiutiu
5
Figure 4. Tone burst signal: (a) time domain and (b) frequency domain (From Giurgiutiu,3 p. 153).
where
Ns ðjÞ ¼ jb j2 þ b2 cos ad cos bd;
2
Ds ¼ j2 b2 cos ad sin bd þ 4j2 ab sin ad cos bd
NA ðjÞ ¼ jb j2 þ b2 sin ad sin bd;
2
DA ¼ j2 b2 sin ad cos bd þ 4j2 ab cos ad sin bd
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
v2
v2
l þ 2m
2
2
2
2
;
a ¼ 2 j ; b ¼ 2 j ; cp ¼
r
cp
cs
rﬃﬃﬃﬃ
m
at 0
ð4Þ
cs ¼
; kPWAS ¼ i
r
m
kPWAS is the complex transduction coefficient that converts applied voltage into guided wave strain at the
T-PWAS, a is half length of PWAS size, and d is plate
half thickness. The modal participation functions SðvÞ
and AðvÞ determine the amplitudes of the S0 and A0
wave modes. The terms sinðjS aÞ and sinðjA aÞ control
the tuning between the PWAS transducer and the
Lamb waves. l and m are Lame’s constants of the
structural material, and r is the material density. The
wavenumber j of a specific mode for certain frequency
v is calculated from Rayleigh–Lamb equation
"
#61
tan bd
4abj2
¼ 2
tan ad
j 2 b2
ð5Þ
where +1 exponent corresponds to symmetric Lamb
wave modes and 21 exponent corresponds to antisymmetric Lamb wave modes.
Step 3. Multiply the structural transfer function by
frequency-domain excitation signal (Figure 4(b)) to
obtain the frequency-domain signal at the R-PWAS,
Figure 5. (a) T-PWAS signal and (b) R-PWAS signal.
T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS:
receiver piezoelectric wafer active sensor.
~R ðxr ; vÞ ¼ Gðxr ; vÞ V
~T ðvÞ. Hence, the wave
that is, V
arriving at the R-PWAS location is
~R ðxr ; vÞ ¼ S ðvÞV
~T ðvÞeijS xr þ AðvÞV
~T ðvÞeijA xr
V
ð6Þ
This signal in equation (6) can be decomposed into
symmetric and antisymmetric components
~ S ðxr ; vÞ ¼ S ðvÞV
~T ðvÞeijS xr
V
R
ð7Þ
~ A ðxr ; vÞ ¼ AðvÞV
~T ðvÞeijA xr
V
R
ð8Þ
Step 4. Perform the inverse Fourier transform to
obtain the time-domain wave signal at the R-PWAS,
that is
VR ðxr ; tÞ ¼ IFFT fV~R ðxr ; vÞg
ð9Þ
Due to the multi-mode character of guided Lamb
wave propagation, the received signal has at least two
separate wave packets, S0 and A0 (Figure 5).
This analysis can be extended to include higher
guided wave modes (S1, A1, etc.), that is
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6
Structural Health Monitoring
Receiver PWAS Signal:
VR ( xd , xr , ω )
Transmitter PWAS
Excitation: VT (ω )
x=0
x = xr
x = xd
Damage
T-PWAS
R-PWAS
Ultrasonic guided
waves from TPiezoelectric
PWAS
transduction
Elec.→Mech Transfer function
.
G ( xd , ω )
Nonlinear damage: new wave source
Transmission, reflection, mode
conversion, and higher harmonics
Ultrasonic waves
with damage
information
Transfer function
G ( xr − xd , ω )
Piezoelectric
transduction
Mech.→Elec.
Figure 6. A pitch-catch configuration between a transmitter PWAS and a receiver PWAS.
PWAS: receiver piezoelectric wafer active sensor.
~R ðxr ; vÞ ¼
V
X
j
~ T ðvÞeij
S ðvÞV
S
xr
S
þ
X
j
~ T ðvÞeij
AðvÞV
A
xr
A
ð10Þ
All the wave modes propagate independently in the
structure. The final waveform will be the superposition
of all the propagating waves and will have the contribution from each Lamb mode.
Insertion of damage effects into the analytical model
Figure 6 shows the pitch-catch active sensing method
for damage detection: the T-PWAS transducer generates ultrasonic guided waves which propagate into the
structure, interact with structural damage at x ¼ xd ,
carry the damage information with them, and are
picked up by the R-PWAS transducer at x ¼ xr .
To model the damage effect on Lamb wave propagation, we consider the damage as a new wave source
at x ¼ xd , and we add mode conversion and nonlinear
sources at the damage location through damage interaction coefficients. The predictive analytical model for
Lamb wave interaction with damage is constructed in
frequency domain in the following steps:
Step 1. This step is identical to step 1 of the pristine
case. Perform Fourier transform of the time-domain
excitation signal VT ðtÞ to obtain the frequency-domain
~T ðvÞ.
excitation spectrum, V
Step 2. Calculate the frequency-domain structural transfer function up to the damage location, Gðxd ; vÞ. The
structure transfer function Gðxd ; vÞ is similar to equation (3) of previous section, only that x ¼ xd , that is
Gðxd ; vÞ ¼ S ðvÞeij
S
xd
þ AðvÞeij
A
xd
ð11Þ
Step 3. Multiply the structural transfer function by
frequency-domain excitation signal to obtain the
frequency-domain signal at the damage location, that
~D ðxd ; vÞ ¼ Gðxd ; vÞ V
~T ðvÞ. Hence, the signal at
is, V
the damage location is
~D ðxd ; vÞ ¼ S ðvÞV
~T ðvÞeijS xd þ AðvÞV
~T ðvÞeijA xd ð12Þ
V
This signal could be decomposed into symmetric and
antisymmetric components
~ S ðxd ; vÞ ¼ S ðvÞV
~T ðvÞeijS xd
V
D
ð13Þ
~ A ðxd ; vÞ ¼ AðvÞV
~T ðvÞeijA xd
V
D
ð14Þ
Step 4. The wave signal at the damage location takes
the damage information by considering transmission,
reflection, mode conversion, and higher harmonics.
Each of these addition phenomena is modeled as a new
wave source at the damage location using damage
interaction coefficients (Figure 7). We distinguish two
damage interaction types: (a) linear and (b) nonlinear,
as discussed next.
Linear damage interaction. Wave transmission, reflection,
and mode conversion are realized by using complexamplitude damage interaction coefficients. Our notations are as follows: we use three letters to describe the
interaction phenomena, with the first letter denoting
the incident wave type, the second letter standing for
resulting wave type, and the third letter meaning propagation direction (transmission/reflection). For instance,
symmetric-symmetric-transmission (SST) means the
incident symmetric waves transmitted as symmetric
waves, while symmetric-antisymmetric-transmission
(SAT) means incident symmetric waves transmitted and
mode converted to antisymmetric waves. Thus, the
complex-amplitude damage interaction coefficient
CSST eiuSST denotes the transmitted symmetric mode
generated by incident symmetric mode with magnitude
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Shen and Giurgiutiu
7
Figure 7. Modeling wave transmission, reflection, mode conversion, and higher harmonics components.
CSST and phase uSST . Similarly, CSAT eiuSAT represents
the transmitted antisymmetric mode generated by incident symmetric mode with magnitude CSAT and phase
uSAT . These coefficients are determined by the features of
the damage and are to be imported into the WFR model.
where M is the number of higher harmonics considered.
For linear interaction with damage, M equals to one.
Step 6. Perform inverse Fourier transform to obtain
the time-domain receiver sensing signal
Nonlinear damage interaction. The center frequency of
waves arriving at the damage location can be obtained
from equations (13) and (14) as vc . The second and
third higher harmonics act as wave sources with center
frequencies of 2vc and 3vc , respectively. Modeling of
higher harmonics is achieved by moving the frequencydomain signal at the damage location to the right-hand
side of the frequency axis by vc and 2vc , that is,
~D ðxd ; v vc Þ
~3D ðxd ; vÞ ¼
~2D ðxd ; vÞ ¼ V
and
V
V
~D ðxd ; v 2vc Þ represent the second and third higher
V
harmonics nonlinear wave source.
The nonlinear damage interaction coefficients are
defined in the same way as the linear ones. For instance,
the complex-amplitude
damage interaction coefficient
M
M
eiuSST denotes the Mth higher harmonics transCSST
mitted symmetric mode generated by incident symM
and phase uM
.
metric mode with magnitude CSST
SST
It should be noted that the above analysis only considers S0 and A0 modes. But the principle could be easily extended to higher modes (S1, A1, etc.). The
difficulty with extending to higher modes will be on
defining the increasing number of transmission, reflection, and mode conversion coefficients. For each
excited Lamb mode, the interaction with damage may
result in more mode conversion possibilities. In this
study, the WFR has been designed to simulate (a)
multi-mode (S0, A0, S1, A1) Lamb waves propagation
in pristine plates and (b) fundamental modes (S0 and
A0) Lamb wave interaction with damage.
Step 5. The guided waves from the new wave sources created at the damage location propagate through the rest
of the structure and arrive at the R-PWAS. The received
wave signal is calculated in frequency domain as
The analytical representation of this process was coded
in MATLAB and resulted in the GUI called WFR
shown in Figure 8.
~ R ðxd ; xr ; vÞ ¼
V
þ
m h
X
M¼1
m h
X
VR ðxd ; xr ; tÞ ¼ IFFT fV~R ðxd ; xr ; vÞg
WFR interface and main functions
i
M iuM
~ S ðxd ; vÞ þ C M eiuMAST V
~ A ðxd ; vÞ eijS ðxr xd Þ
CSST
e SST V
MD
AST
MD
M
M
CAAT
eiuAAT
ð16Þ
i
~ A ðxd ; vÞ þ C M eiuMSAT V
~ S ðxd ; vÞ eijA ðxr xd Þ
V
MD
SAT
MD
M¼1
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ð15Þ
8
Structural Health Monitoring
Figure 8. Main GUI of WaveFormRevealer.
GUI: graphic user interface.
WFR allows users to control several parameters:
structure material properties, PWAS size, location of
sensors, location of damage, damage type (linear/nonlinear damage of various severities), and excitation signal (frequencies, count numbers, signal mode
excitation, arbitrary waveform type, etc.). Dual display
of waveforms allows for the sensing signals to be shown
at two different sensor locations. For instance, Figure 8
shows two receiver waveforms at locations x1 ¼ 0 mm
and x2 ¼ 500 mm as measured from the transmitter (in
this case x1 ¼ 0 mm means that R-PWAS-1 collocated
with the T-PWAS). Thus, PWAS-1 shows the reflections from damage, and PWAS-2 shows the signal
modified after passing through the damage. Users are
able to conduct fast parametric studies with WFR. It
may take several hours for commercial finite element
software to obtain an acceptable-accuracy solution for
high-frequency, long distance propagating waves, but it
takes only several seconds to obtain the same solution
with WFR. Besides analytical waveform solutions, the
WFR can also provide users with wave speed
dispersion curves, tuning curves, frequency components
of received wave packets, structure transfer function,
and so on. All the calculated results are fully available
to the user, and could be saved in Excel files by clicking
the ‘‘SAVE’’ button. Figure 9 shows a case study for
Lamb wave propagation of a 100 kHz tone burst in a
1-mm-thick aluminum plate. Figure 9(a) shows the dispersion curves; Figure 9(b) shows the excitation spectrum overlap with the S0 and A0 tuning curves. Figure
9(c) shows the spectra of the S0 and A0 packets displaying frequency shifts to the right and to the left,
respectively, due to the interaction between excitation
spectrum and the tuning curves. Figure 9(d) shows the
structure transfer function Gðx; vÞ.
Besides the main interface, WFR has two subinterfaces shown in Figure 10: (a) damage information
platform and (b) guided wave spatial propagation solver. The damage information platform allows users to
input the damage location and damage interaction
coefficients. For example, SST represents the magnitude of transmitted S0 mode generated by an incoming
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Shen and Giurgiutiu
(a)
9
4
(b)
x 10
2
c (m/s)
A3
S3
Normalized Amplitude
A2
A1
1.5
S1
S2
1
S0
0.5
A0
0
6000
4000
f (kHz)
2000
1.2
8000
1
0.8
S0
0.6
0.4
A0
0.2
0
10000
0
200
600
400
Frequency kHz
800
1000
100
200
300
Frequency kHz
400
500
(d)
1.5
Excitation 100 kHz
1
0.8
A0
0.6
Magnitude
Normalized Amplitude
(c)
0
1.2 Excitation
S0
0.4
1
0.5
0.2
0
0
50
100
150
Frequency kHz
200
0
0
Figure 9. Calculation of various quantities in Lamb wave propagation: (a) wave speed dispersion curve, (b) tuning curve, (c)
frequency contents of received wave packets, and (d) structure transfer function.
S0 mode, whereas SAT and phi-SAT represent the magnitude and phase of the transmitted A0 mode resulting
from the mode conversion of an incoming S0 mode. The
values of these damage interaction coefficients are not
calculated by the WFR. This gives the users the freedom
to define their own specific problem. For instance, a particular type of damage (plastic zone, fatigue, cracks)
with certain degree of severity will have different interaction characteristics with the interrogating guided waves.
These coefficients may be determined experimentally or
calculated through other methods (analytical, FEM,
BEM, etc.). Among all the above methods, FEM
approach shows good results for obtaining the interaction coefficients of arbitrary shaped damage. Successful
examples and details can be found in Velichko and
Wilcox20,21 and Moreau et al.22 In an example presented
later in this study, we used a trial-and-error approach to
tune the WFR coefficients to the data obtained from
experiments and finite element simulations.
The spatial propagation solver is like a B-scan. Using
the analytical procedure, we obtain the time-domain
waveform solution at various locations along the structure. Thus, the time-domain waveform solutions of a
sequence of points along the wave propagation path are
obtained. If we select the sequence of solution points
fine enough, a time-spatial domain solution of the wave
field is obtained. The spatial solution of wave field at a
particular instance in time is available as shown in
Figure 10(b). After the time-spatial solution of wave
field is obtained, we can do the frequency–wavenumber
analysis23 to see the wave components of the signal
(Figure 11). These will be illustrated in the case studies
discussed later in this article.
Case studies
Linear interaction with damage of selective Lamb
wave modes
WFR allows users to select single mode (S0 and A0) or
multi-mode (S0 and A0) to be excited into the structure. Three test cases were conducted: (a) incident S0
wave linear interaction with damage, (b) incident A0
wave linear interaction with damage, and (c) combined
S0 and A0 waves linear interaction with damage. The
test case setup is shown in Figure 12. The T-PWAS and
R-PWAS are placed 600 mm away from each other on
a 1-mm-thick aluminum 2024-T3 plate. The damage is
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10
Structural Health Monitoring
Figure 10. User interfaces: (a) damage information platform and (b) guided wave spatial propagation solver.
placed 200 mm from the T-PWAS. A 5-count Hanning
window modulated tone burst centered at 100 kHz is
used as the excitation. The time-domain and the time–
frequency domain signals of the test cases are shown in
Figure 13.
Figure 13 shows that new wave packets appear due
to the interaction between interrogation Lamb waves
and damage. Incident S0 wave will generate A0 wave
from mode conversion at the damage, whereas incident
A0 wave will generate S0 wave from mode conversion
at the damage. However, from the time-frequency analysis, it could be observed that after linear interaction,
the frequency spectrum of the waves still center around
the excitation frequency 100 kHz.
Nonlinear interaction with damage of selective Lamb
wave modes
As test cases for nonlinear interaction between Lamb
waves and damage, three simulations were carried out:
Figure 11. Frequency–wavenumber display window.
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Shen and Giurgiutiu
11
T-PWAS
R-PWAS
Damage
200 mm
400 mm
Figure 12. Test case setup for pitch-catch Lamb wave interaction with damage.
T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor.
Time domain signal from WFR
0.5
0
-0.5
Transmitted S0
-1
50
100
150
400
300
200
100
200
250
300
350
400
450
500
0
50
100
150
Time domain signal from WFR
1
A0 Excitation
0.5
Mode converted S0
from incident A0 wave
-0.5
Transmitted A0
-1
100
(c)
150
200
250
300
350
400
450
0
Transmitted S0
50
100
New packet from mode
conversion at damage
150
200
250
300
500
400
450
500
400
450
500
200
100
0
50
100
350
150
200
250
300
350
Transmitted A0
500
400
300
200
100
0
0
450
Time (microsecond)
Time-frequency domain signal
0.5
-1
400
300
500
S0 and A0 Excitation
-0.5
350
400
Time (microsecond)
Time domain signal from WFR
1
300
500
0
50
250
Time-frequency domain signal
0
0
200
Time (microsecond)
Time (microsecond)
Frequency (kHz)
Normalized amplitude
500
0
0
(b)
Normalized amplitude
Time-frequency domain signal
Mode converted A0
from incident S0 wave
S0 Excitation
Frequency (kHz)
1
Frequency (kHz)
Normalized amplitude
(a)
400
450
500
Time (microsecond)
0
50
100
150
200
250
300
350
Time (microsecond)
Figure 13. Simulation of linear interaction between Lamb waves and damage: (a) S0 mode excitation, (b) A0 mode excitation, and
(c) S0 and A0 modes excitation. It should be noted that no higher harmonics are observed.
WFR: WaveFormRevealer.
(a) incident S0 wave nonlinear interaction with damage,
(b) incident A0 wave nonlinear interaction with damage, and (c) combined S0 and A0 waves nonlinear interaction with damage. The test case setup is the same as
shown in Figure 12, only the interaction with damage is
nonlinear. The time signals and the time-frequency
analysis of the test cases are shown in Figure 14.
It can be observed in Figure 14 that after nonlinear
interaction with the damage, the waveforms become
distorted and contain distinctive nonlinear higher harmonics. For S0 waves which are less dispersive at the
given frequency range, the nonlinear higher harmonics
stay inside the wave packet. However, for A0 waves
which are dispersive at the given frequency range, the
higher harmonic components travel faster, leading the
way and may escape from the fundamental wave
packet.
Experimental verifications
Multi-mode Lamb wave propagation in a pristine
plate
In our study, two PWAS transducers were mounted on
a 3.17-mm-thick aluminum 7075-T6 plate. Figure 15
shows the experiment setup. The T-PWAS sends out
ultrasonic guided waves into the structure. The guided
waves, that is, Lamb waves propagate in the plate,
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12
Structural Health Monitoring
Time domain signal from WFR
S0 Excitation
0
-0.5
Transmitted S0
with distortion
-1
0
50
100
150
200
500
300
350
400
450
Dispersive A0 mode
generated from mode
conversion from S0 wave
400
300
200
100
0
250
500
0
50
100
1
A0 Excitation
0
Mode converted S0 with
distortion from incident A0
-1
0
50
100
150
200
250
Transmitted A0
350
1
400
450
300
0
50
100
150
200
250
300
Transmitted A0
350
450
500
nonlinear
higher harmonics
100
0
50
100
150
200
250
300
350
400
450
500
400
450
500
Time-frequency domain signal
0
-1
400
Time (microsecond)
0.5
Transmitted S0 New packet from mode
with distortion conversion at damage
350
200 Distinctive
500
S0 and A0 Excitation
-0.5
300
Dispersive transmitted
A0 wave
400
Time (microsecond)
Time domain signal from WFR
(c)
250
500
0
300
200
Time-frequency domain signal
0.5
-0.5
150
Time (microsecond)
Time (microsecond)
Time domain signal from WFR
Frequency (kHz)
Normalized amplitude
Mode converted A0
from incident S0 wave
0.5
(b)
Normalized amplitude
Time-frequency domain signal
Frequency (kHz)
1
400
450
500
Frequency (kHz)
Normalized amplitude
(a)
500
400
300
200
100
0
0
50
100
150
200
250
300
350
Time (microsecond)
Time (microsecond)
Figure 14. Simulation of nonlinear interaction between Lamb waves and damage: (a) S0 mode excitation, (b) A0 mode excitation,
and (c) S0 and A0 modes excitation. It should be noted that distinctive higher harmonics are observed.
WFR: WaveFormRevealer.
T-PWAS
303 mm propagation path
R-PWAS
Figure 15. Experiment setup for multi-mode Lamb wave
propagation.
WFR: WaveFormRevealer.
undergoing dispersion and are picked up by the
R-PWAS. The Lamb waves are multi-modal, hence
several wave packets appear in the received signal.
Agilent 33120A Arbitrary Waveform Generator is used
to generate 3-count Hanning window modulated tone
burst excitations. A Tektronix Digital Oscilloscope is
used to record the experimental waveforms. The excitation frequency is increased from 300 to 600 kHz.
Corresponding plate material, thickness, PWAS size,
and sensing location information is input into the
WFR. The analytical waveforms of various frequencies
are obtained. Figure 16 shows the comparison between
analytical solution from WFR and experimental data.
It can be observed that at 300 kHz, only S0 and A0
modes exist. The WFR solution matches well with
experimental data. At 450 kHz, S0 mode becomes more
dispersive; besides S0 and A0 modes, A1 mode starts to
pick up with highly dispersive feature. At 600 kHz, S0,
A0, and A1 modes exist simultaneously. The simulation results and the experimental data have slight differences due to the fact that 1D analytical formulas
and pin force excitation assumptions are used in this
study. To further validate WFR predictions, we also
conducted 2D FEM simulation with pin force excitation (1D Lamb wave propagation simulation). Figure
17 shows the comparison between WFR and FEM
simulations. It can be observed that the 300 and
450 kHz waveforms match very well between WFR
and FEM. Signals of 600 kHz also have reasonably
good agreement. It should be noted, even for 1D Lamb
wave propagation simulation, that the 600 kHz wave
computation requires considerably small element size
and time marching step. The FEM simulation for such
high-frequency, short-wavelength situation is becoming
prohibitive due to the heavy consumption of computation time and computer resources. On the contrary,
WFR only requires several seconds to obtain the same
results due to its highly efficient analytical formulation.
The guided wave spatial propagation solver in WFR
is used to obtain the time–space wave field (B-scan) as
shown in Figure 18(a). The frequency–wavenumber
analysis is conducted next, as shown in Figure 18(b).
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Shen and Giurgiutiu
13
1
0.5
A0
S0
300 kHz
Experiment
WFR
0
-0.5
Normalized Amplitude
-1
0
20
40
60
80
100
120
140
1
160
180
Experiment
WFR
0.5
450 kHz
0
-0.5
S0 dispersive wave
0
20
40
A1 dispersive wave
A0
-1
60
80
100
120
140
1
160
180
Experiment
WFR
0.5
600 kHz
0
-0.5
S0 dispersive wave
A0
-1
0
20
40
60
80
100
A1 dispersive wave
120
140
160
180
Time (microsecond)
Figure 16. Comparison between WFR and experiment for multi-mode Lamb wave propagation in a pristine 3.17-mm aluminum
plate.
WFR: WaveFormRevealer.
The 600 kHz case is used as an example. From the Bscan, S0, A0, and A1 wave components can be
observed. Frequency–wavenumber analysis gives very
clear information on the wave mode components of the
wave field. Transmitted S0 wave (S0-T), A0 wave
(A0-T), and A1 wave (A1-T) are clearly noticed in
Figure 18(b).
Linear interaction between Lamb waves and damage
Pitch-catch mode. Figure 19 shows the experimental specimen (3.17-mm-thick Aluminum-7075-T6 plate), with
PWAS #3 used as the transmitter (T-PWAS) and
PWAS #4 used as the receiver (R-PWAS). A notch
(h1 ¼ 2:5 mm; d1 ¼ 0:25 mm) is machined on the plate,
143.5 mm from the T-PWAS. The wave propagation
path from T-PWAS to R-PWAS is 303 mm. The 3count Hanning window modulated tone burst signals
with center frequencies varying from 150 to 300 kHz
are used as the excitation.
S0 and A0 waves are transmitted by the T-PWAS.
At the notch, S0 waves will be transmitted as S0 waves
and also will be mode converted to transmitted A0
waves. A0 waves will be transmitted as A0 waves and
also will be mode converted to transmitted S0 waves.
All these transmitted waves will propagate along the
rest of the structure and be picked up by the R-PWAS.
The damage interaction coefficients are physically
determined by the size, severity, type of the damage. In
this study, we used a trial-and-error approach to tune
the WFR damage interaction coefficients to the data
obtained from the experiments. The adjusted damage
interaction coefficients which gave best match with
experiments for 150 kHz excitation case are shown in
Table 1.
Figure 20 shows the WFR simulation results compared with experiments. It can be noticed that the analytical waveforms agree well with experimental data. A
new wave packet is generated due to mode conversion
at the notch.
Pulse-echo mode. Figure 21 shows the experimental
setup for pulse-echo active sensing method. The same
specimen is used, with an R-PWAS bounded side by
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14
Structural Health Monitoring
1
0.5
WFR
FEM
300 kHz
0
-0.5
Normalized Amplitude
-1
0
20
40
60
80
100
120
140
160
1
0.5
180
WFR
FEM
450 kHz
0
-0.5
-1
0
20
40
60
80
100
120
140
160
1
0.5
180
WFR
FEM
600 kHz
0
-0.5
-1
0
20
40
60
80
100
120
140
160
180
Time (microsecond)
Figure 17. Comparison between WFR and FEM for multi-mode Lamb wave propagation in a pristine 3.17-mm aluminum plate.
WFR: WaveFormRevealer; FEM: finite element method.
(a)
(b)
Time-space
wave
fieldSignal
(B-scan)
Time-Space
Domain
Frequency-wavenumber
analysis
Frequency-Wavenumber
Domain
Signal
2000
300
A0-T
250
1000
200
150
Wavenumber
Space (mm)
A1
S0
A0
100
S0-T
A1-T
0
A1-R
S0-R
-1000
50
0
S1-T
S1-R
A0-R
-2000
0
20
40
60
80
100
Time (microsecond)
120
0
500
1000
1500
Frequency (kHz)
2000
Figure 18. (a) Time–space wave field (B-scan) and (b) frequency–wavenumber analysis from WFR.
WFR: WaveFormRevealer.
side to the T-PWAS. The 3-count Hanning window
modulated tone burst signals with the center frequency
of 95.5 kHz is used as the excitation. Guided Lamb
waves generated by the T-PWAS will propagate into
the structure, reach the notch, and be reflected back as
echoes. At the notch, S0 waves will be reflected as S0
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Shen and Giurgiutiu
15
Figure 19. Experiment for Lamb wave linear interaction with a notch (pitch-catch mode).
T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor.
1
Experiment
WFR
S0
0.5
A0
150 kHz
0
-0.5
New packet from mode conversion (S0+A0)
-1
0
1
Normalized Amplitude
0.5
20
40
60
80
100
120
140
160
180
Experiment
WFR
200 kHz
0
-0.5
New packet from mode conversion (S0+A0)
-1
0
1
0.5
20
40
60
80
100
120
140
160
180
60
80
100
120
140
160
180
60
80
100
120
140
160
180
Experiment
WFR
250 kHz
0
-0.5
-1
0
1
20
40
Experiment
WFR
0.5
300 kHz
0
-0.5
-1
0
20
40
Time (microsecond)
Figure 20. Comparison between WFR simulations and experiments for Lamb wave interaction with a notch in pitch-catch mode.
WFR: WaveFormRevealer.
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16
Structural Health Monitoring
T-PWAS
143.5 mm
Notch
R-PWAS
Figure 21. Experiment for Lamb wave linear interaction with a
notch (pulse-echo mode).
T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS:
receiver piezoelectric wafer active sensor.
waves and also will be mode converted to reflected A0
waves. A0 waves will be reflected as A0 waves and also
will be mode converted to reflected S0 waves. All the
echoes will reach the R-PWAS and be picked up.
The adjusted damage interaction coefficients which
gave best match with the experiment are shown in
Table 2.
Figure 22 shows the WFR simulation result compared with the experiment. The reflected S0 and A0
wave packets could be observed. The new waves
between S0 and A0 wave packets are from mode
Table 1. Damage interaction coefficients for pitch-catch mode.
1
CSST
0.55
u1SST
230
Magnitude coefficient
Value (normalized)
Phase coefficient
Value (°)
1
CSAT
0.11
u1SAT
30
1
CAAT
0.8
u1AAT
0
1
CAST
0.06
u1AST
30
Table 2. Damage interaction coefficients for pulse-echo mode.
1
CSSR
0.2
u1SSR
60
Normalized amplitude
Magnitude coefficient
Value (normalized)
Phase coefficient
Value (°)
1
CSAR
0.04
u1SAR
60
1
CAAR
0.12
u1AAR
260
Direct waves
0.2
1
CASR
0.04
u1ASR
60
ReflectedS0
conversion at the notch. The analytical simulation
matches the experiment data. Differences are noticed:
first, the direct waves have a phase shift due to the fact
that the R-PWAS and T-PWAS are some distance
away from each other, while in our analytical model,
we consider them to be at the same location; second,
the boundary reflections are present and mixed with
the weak echoes from the notch in the experiment, but
in our model, the boundary reflections are not
considered.
Figure 23 shows the results from WFR spatial propagation solver. The wave transmission, reflection, and
mode conversion can be clearly noticed in both the Bscan and frequency–wavenumber analysis. It is apparent that the wave field contains transmitted S0 and A0
modes, and reflected S0 and A0 modes.
Nonlinear interaction between Lamb waves and
damage
A guided wave pitch-catch method may be used to
interrogate a plate with a breathing crack which opens
and closes under tension and compression.6,24 The
ultrasonic waves generated by the T-PWAS propagate
into the structure, interact with the breathing crack,
acquire nonlinear features, and are picked up by the RPWAS. This process is shown in Figure 24. The nonlinear interaction between Lamb waves and the breathing crack will introduce nonlinear higher harmonics
into the interrogation waves. A multi-physics transient
finite element model was used to simulate the Lamb
wave interaction with a nonlinear breathing crack. The
damage interaction coefficients obtained from fitting
the FEM solution (Table 3) were input into the WFR
simulator.
Figure 25 shows the comparison between FEM and
the WFR analytical solution. It is noticed that the
FEM results and the analytical solution agree very well
because the damage interaction coefficients were fitted
to the FEM solution. The time-domain waveforms
show nonlinear characteristics of noticeable nonlinear
ReflectedA0
Experiment
WFR
0.1
0
-0.1
-0.2
Wave packet from mode conversion
0
50
100
150
Boundary reflections
200
250
300
Time (microsecond)
Figure 22. Comparison between WFR simulations and experiments for Lamb wave interaction with a notch in pulse-echo mode.
WFR: WaveFormRevealer.
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Shen and Giurgiutiu
(a)
17
(b)
300
AST
AAT
250
1500
Forward transmission
through damage
1000
A0-T
Wavenumber
Space (mm)
SST
200
SAT
150 Damage
100
500
S0-T
0
-500
S0-R
A0-R
SAR
50
ASR
SSR
0
0
-1000
AAR
-1500
50
100
Time (microsecond)
Backward reflection
from damage
0
100
200
300
400
Frequency (kHz)
500
Figure 23. (a) Time–space domain solution (B-scan) and (b) frequency–wavenumber analysis from WFR with transmission,
reflection, and mode conversion damage effects.
WFR: WaveFormRevealer.
Generation of
higher harmonics
……..
Breathing crack
T-PWAS
R-PWAS
Figure 24. Pitch-catch method for the detection of breathing crack; the mode conversion at the crack is illustrated by the two
arrows.
T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor.
Table 3. Nonlinear interaction coefficients.
Magnitude coefficient
Value (normalized)
Phase coefficient
Value (°)
1
CSST
0.900
u1SST
0
1
CSAT
0.420
u1SAT
100
1
CAAT
0.820
u1AAT
235
1
CAST
0.100
u1AST
90
2
CSST
0.082
u2SST
0
distortion in S0 packet and zigzags in the new packet.
The frequency spectrums show distinctive nonlinear
higher harmonics (200 and 300 kHz). Since we only
consider up to the third higher harmonic in this case
study, the frequency domain of analytical solution
shows only the first three peaks, while the finite element
solution have even higher harmonics. But the solution
up to the third higher harmonics is accurate enough to
render an acceptable waveform in time domain.
The guided wave spatial propagation solver in WFR
was used to obtain the time–space wave field. Figure 26
shows the time–space wave field and frequency–
wavenumber analysis of Lamb wave interaction with
nonlinear breathing crack.
2
CSAT
0.100
u2SAT
0
2
CAAT
0.050
u2AAT
120
2
CAST
0.110
u2AST
90
3
CSST
0.032
u3SST
0
3
CSAT
0.038
u3SAT
0
3
CAAT
0.005
u3AAT
0
3
CAST
0.025
u3AST
0
Transmission, reflection, and mode conversion phenomena at the damage can be clearly noticed. The
frequency–wavenumber analysis reveals the wave components during the interaction process. The wave field
contains transmitted S0 and A0 waves and reflected S0
and A0 waves. Nonlinear higher harmonics can be
observed at 200 kHz.
The WFR-guided wave spatial propagation solver
can provide the spatial wave pattern at any instance of
time. The spatial waveforms at 0, 25, 50, 75, 100, 125,
150, 175, and 200 ms are displayed in Figure 27. The
spatial waveforms shows (a) Lamb waves propagating
into the structure at T ¼ 25 ms, (b) Lamb modes separating into distinct packets at T ¼ 50 ms, (c) Lamb wave
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18
Structural Health Monitoring
(a)
1
Normalized amplitude
Waveform with zigzags
A0
0.5
WFR
0
-0.5
New wave packet from
nonlinear interaction
Distorted S0
FEM
-1
0
50
100
150
200
250
300
350
Time (microsecond)
(b)
(d)
FEM
0
FEM
FEM
0
10
0
10
Magnitude
10
(c)
-5
10
10
-5
WFR
WFR
0
200
400
600
800
1000
0
200
400
600
WFR
1000 0
800
200
600
800
1000
Frequency (kHz)
Frequency (kHz)
Frequency (kHz)
400
Figure 25. (a) Comparison between finite element simulation (FEM) and analytical simulation (WFR), (b) frequency spectrum of S0
packet, (c) frequency spectrum of new packet and (d) frequency spectrum of A0 packet.
WFR: WaveFormRevealer; FEM: finite element method.
(a)
(b)
500
Nonlinear higher
harmonics
Forward transmission
1000
through damage
300
Damage
200
Wave Number
Space (mm)
400
A0-T
500
S0-T
0
S0-R
-500
A0-R
Backward reflection
-1000
from damage
100
0
1500
0
50
100
150
200
Time (microsecond)
250
300
-1500
0
50
100 150 200 250
Frequency (kHz)
Figure 26. (a) Time–space wave field (B-scan) and (b) frequency–wavenumber analysis from WFR.
WFR: WaveFormRevealer.
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300
350
Shen and Giurgiutiu
19
1
T = 0 μs
Damage
↓
0
-1
0
50
100
150
1
200
250
300
350
400
450
500
T = 25 μs
Damage
↓
0
Waves propagating into the structure; S0 and A0 mixed together
-1
0
1
50
100
A0
150
S0
0
0
50
100
150
1
250
300
400
450
500
T = 50 μs
↓
S0 wave interacting with damage
200
250
300
350
400
↓
450
500
T = 75 μs
Transmitted S0
Damage
0
350
Damage
S0 and A0 model separation
-1
Normalized Amplitude
200
Transmitted A0 from mode conversion
-1
0
1
50
100
150
Reflected A0 from
mode conversion
0
Reflected S0
-1
0
50
250
Damage
100
150
0
50
100
150
200
250
100
300
350
400
450
500
T = 125 μs
New wave packet from mode conversion
containing both S0 and A0 modes
200
250
300
350
400
450
500
T = 150 μs
↓
Reflected A0
50
500
Transmitted S0
Damage
0
0
450
T = 100 μs
↓
Reflected S0 from
mode conversion
-1
400
↓
Reflected S0 from
mode conversion
1
350
Damage
0
-1
300
A0 wave interacting with damage
New wave packet from mode conversion
Reflected A0 from
mode conversion
1
200
150
1
Transmitted A0
200
250
300
350
400
450
500
T = 200 μs
Damage
↓
0
Transmitted A0
Reflected A0
-1
0
50
100
150
200
250
300
Space (mm)
350
400
Figure 27. Spatial wave propagation of Lamb wave interaction with breathing crack (calculated using WFR).
WFR: WaveFormRevealer.
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450
500
20
Structural Health Monitoring
packets interaction with the damage also at T ¼ 50 ms,
and (d) wave transmission, reflection, mode conversion,
and nonlinear distortion of waveforms at various
instances (T ¼ 75; 100; 125; 150; and 200 ms).
Summary, conclusions, and future work
Summary
should be built to simulate wave attenuation in waveguides. Boundary reflection and damage effects in 2D
wave propagation should be investigated. Attempts for
simulating guided wave propagation in composite
structures should be made using WFR.
Declaration of conflicting interests
The authors declare that there is no conflict of interest.
In this study, we presented the WFR—an analytical
framework and predictive tool for the simulation of
guided Lamb wave interaction with damage. The theory of inserting damage effects into the analytical
model was addressed, including wave transmission,
reflection, mode conversion, and nonlinear higher harmonics components. The analytical model was coded
into MATLAB, and the WFR GUI was developed to
obtain fast predictive waveforms for arbitrary combinations of sensors, structural properties, and damage.
Main functions of WFR were introduced, including the
calculation for dispersion curves, tuning curves, frequency spectrum of sensing signal, plate transfer function, time–space domain waveforms with damage
effects, frequency–wavenumber analysis, and the capability of considering arbitrary user defined excitation
signals. Test cases were carried out. Experimental verifications were presented. The predictive solution from
WFR agreed well with experiments and finite element
simulations. WFR can be downloaded from: http://
www.me.sc.edu/Research/lamss/html/software.html.
Conclusion
The WFR was capable of calculating dispersion curves,
tuning curves, frequency components of wave packets,
and structural transfer function. It could be used to
obtain time–space domain waveforms with damage
effects and frequency–wavenumber analysis. WFR
could provide fast predictive solutions for multi-mode
Lamb wave propagation and interaction with linear/
nonlinear damage. The solutions compared well with
experiments and finite element simulations. It was also
found that computational time savings of several orders
of magnitude are obtained by using the analytical
model WFR instead of FEM methods. WFR allowed
users to conduct fast parametric studies with their own
designed materials, geometries, and excitations.
Future work
Rational methods for determining damage interaction
coefficient values need to be found (not trial and error).
Work should be carried out to extend the analysis to
2D wave propagation (three-dimensional (3D) FEM
and 2D WFR). The 2D WFR with damping effect
Funding
The following funding supports for this study are thankfully
acknowledged: Office of Naval Research # N00014-11-10271, Dr. Ignacio Perez, Technical Representative; Air Force
Office of Scientific Research # FA9550-11-1-0133, Dr. David
Stargel, Program Manager.
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